Abstract
Bours [4] recently showed some constructions for perfect 2 and 3-deletion-correcting codes from combinatorial designs. He settled existence of perfect 2-deletion-correcting codes with words of length 4. However, the existence of perfect 3-deletion-correcting codes with words of length 5, or T*(2, 5, v), remained unsettled for v ≡ 7, 8 (mod 10) and v = 13, 14, 15, 16. In this paper we provide new constructions for these codes from combinatorial designs, and show that a T*(2, 5, v) exists for all v.
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A. Assaf, N. Shalaby, A. Mahmoodi, and J. Yin, Directed packings with block size 5 and odd v, Australasian Journal of Combinatorics, Vol. 13 (1996) pp. 217-226.
F. E. Bennett, R. Wei, J. Yin, and A. Mahmoodi, Existence of DBIBDs with block size six, Utilitas Mathematica, Vol. 43 (1993), pp. 205-217.
Th. Beth, D. Jungnickel, and H. Lenz, Design Theory, Bibliographisches Institut, Zurich (1985).
P. A. H. Bours, On the construction of perfect deletion-correcting codes using design theory, Designs, Codes and Cryptography, Vol. 6 (1995) pp. 5-20.
C. J. Colbourn and A. Rosa, Directed and Mendelsohn triple systems, (Jeffrey H. Dinitz and Douglas R. Stinson, eds.), Contemporary Design Theory: A Collection of Surveys, John Wiley & Sons, Inc. (1992), pp. 97-136.
H. Hanani, Balanced incomplete block designs and related designs. Discrete Math., Vol. 11 (1975) pp. 255-369.
S. H. Y. Hung and N. S. Mendelsohn, Directed triple systems, J. Combin. Theory A, Vol. 14 (1973) pp. 310-318.
V. I. Levenshtein, On perfect codes in deletion and insertion metric, Discrete Math. Appl., Vol. 2, No.3 (1992) pp. 241-258.
R. C. Mullin and J. Yin, On packings of pairs by quintuples v ≡ 3;9 or 17 (mod 20), Ars Combinatoria, Vol. 6 (1993) pp. 161-171.
J. R. Seberry and D. Skillicorn, All directed BIBDs with k = 3 exist, J. Combin. Theory A, Vol. 29 (1980) pp. 244-248.
N. Shalaby and J. Yin, Directed packings with block size 5 and even v, Designs, Codes and Cryptography, Vol. 6 (1995) pp. 133-142.
D. B. Skillicorn. Directed Packings and Coverings with Computer Applications, PhD thesis, University of Manitoba (1981).
D. J. Street and J. R. Seberry, All DBIBDs with block size four exist, Utilitas Math., Vol. 18 (1980) pp. 27-34.
D. J. Street and W. H. Wilson, On directed balanced incomplete block designs with block size five, Utilitas Math., Vol. 18 (1980) pp. 161-174.
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Mahmoodi, A. Existence of Perfect 3-Deletion-Correcting Codes. Designs, Codes and Cryptography 14, 81–87 (1998). https://doi.org/10.1023/A:1008212622423
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DOI: https://doi.org/10.1023/A:1008212622423